Lakshmanan,V., 2012: Image processing of weather radar reflectivity data: Should it be done in Z or dBZ?,Electronic J. Severe Storms Meteor., 7 (3), 1–8.

Image Processing of Weather Radar Reflectivity Data: Should It be Donein Z or dBZ?

VALLIAPPA LAKSHMANANCooperative Institute of Mesoscale Meteorological Studies,

University of Oklahoma & National Severe Storms Laboratory,Norman, Oklahoma

(Submitted 10 January 2012; in final form 24 April 2012)

ABSTRACT

It appears to be a common belief that processing, such as interpolation, clustering or smoothing, of weatherradar reflectivity fields ought to be carried out on the reflectivity factor (Z) and not on its logarithm (dBZ). It isdemonstrated here by means of a statistical study on a large dataset that, contrary to common belief, processing indBZ is better for such applications.

1. Motivation

Image processing of weather radar reflectivityfields (the postprocessing of radar data using valuesat adjacent range gates) traditionally has been carriedout in dBZ, e.g: Delanoy and Troxel (1993); Johnsonet al. (1998); Lakshmanan (2004); Charalampidiset al. (2002); Anagnostou (2004); Steiner and Smith(2002); Kessinger et al. (2003). This author couldfind no successful postprocessing of radar momentdata (not pulses at a single gate, but values at adjacentgates) that has been carried out in terms of thereflectivity factor measured in mmi6 mm−3, hereafterZ.

However, there appears to be a persistent belief,particularly among radar engineers (and noticed bythis author in the course of having many papersreviewed), that image processing of radar reflectivityin an “artificial” quantity such as dBZ is wrong.The radar measures Z, the argument seems to go,so any interpolation or smoothing ought to be carried

Corresponding author address: ValliappaLakshmanan, University of Oklahoma, 120David L. Boren Blvd., Norman OK, E-mail:lakshman@ou.edu

out in Z and not dBZ1. This argument is a fallacy.To see why, imagine a hypothetical instrument thatmeasures the spatial variation of the square root of thedensity of a substance. Suppose that an intermediatepixel was not measured and one must estimate thevalue at the missing pixel. Should the value at themissing pixel be estimated by linearly interpolatingmeasured values (the square root of the densities)at adjacent pixels or the “artificial” values (obtainedby computing the squares of the measurements atthe adjacent pixels, interpolating the squares andthen taking the square root)? Naturally, the answerdepends on the variation of density of the substancebeing measured, on whether the density or the squareroot of the density is linear. The better field forcarrying out linear interpolation has nothing to dowith which quantity gets measured and everythingto do with which quantity varies linearly within thesubstance being measured.

As will be shown later, linear interpolation isclosely related to operations such as clustering

1Note that this paper deals with image processingof radar moment fields, and not with pulses at a singlerange gate. I have nothing to say here about therelative merits of doing signal processing in Z or indBZ.

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and smoothing of radar reflectivity images.Consequently, an appropriate way to address thequestion of whether it is better to carry out imageprocessing operations in Z or in dBZ is to examinea large weather radar reflectivity dataset and verifywhich is more linear: variation in Z or in dBZ.The author carried out such an experiment and reportsthose results in this note.

Does it matter whether one interpolates dBZ orZ? Yes, because the results can be dramaticallydifferent. Imagine interpolating between two values46 dBZ and 53 dBZ. Interpolating in Z wouldgive the mid-way value as 50.8 dBZ whereasinterpolating in dBZ would give the mid-way valueas 49.5 dBZ. This difference of 1.3 dB, especiallyaloft, can be dramatic in its effect on algorithmssuch as hail estimation. Interpolated Z values areconsistently larger than interpolated dBZ values sincean interpolated dBZ value is akin to a geometric meanof the Z values2 and the geometric mean is alwayssmaller than the arithmetic mean for non-negative realnumbers.

2. Method

WSR-88D reflectivity data from the Little Rock,Arkansas radar from 1 January to 31 December 2007(at 1 km × 0.95o resolution)were analyzed in thisexperiment. These data were chosen solely becausethe author happened to have them readily availableon disk. Radials of the lowest elevation scan within230km of the radar were searched for “triads’. Atriad is defined as a group of three adjacent rangegates that all had valid data, i.e. data above thesignal-to-noise threshold.

Within each triad, an interpolated value wascomputed from the two end values and comparedto the value at the center. The interpolation wascarried out in two ways: 1) by simply averagingthe dBZ values and 2) by converting the dBZvalues to Z, averaging the Z values, and thenconverting the averaged Z value back to dBZ. Using

2 This is because:

dBZmean = 0.5(dBZ1 + dBZ2)= 0.5(10log(Z1) + 10log(Z2))= 5log(Z1Z2)= 10log(

√Z1Z2)

this procedure, one can measure the interpolationerror when interpolating in dBZ and in Z. Ifthe interpolation error is measured in dB, theinterpolation error when interpolating in dBZ is givenby:

edBZ(dB) = z0 − (z−1 + z1)/2 (1)

where the reflectivity values in the triad are(z−1, z0, z1) with z0 being the reflectivity value of thecenter gate in dBZ. The interpolation error (in dB)when interpolating in Z is given by:

eZ(dB) = z0 − 10 log10((10z−1/10 + 10z1/10)/2)(2)

Similarly, one can compute the interpolation error inmm6i mm−3 by first converting dBZ values to Z andthen computing the error. The interpolation error inmm6 mm−3 when interpolating in Z is given by:

edBZ(Z) = 10z0/10 − 10(z−1+z1)/20 (3)

whereas when interpolating in dBZ is given by:

eZ(Z) = 10z0/10 − (10z−1/10 + 10z1/10)/2 (4)

3. Results

Statistics of the interpolation errors on all triadsfrom a year’s set of WSR-88D data are shownin Table 1. The root mean square error (RMSE)is shown first with errors in dB and then inmm6mm−3. As can be observed readily from thefirst row of that table, interpolation using dBZ isbetter whether we consider the RMSE, the bias,the variance or a pair-wise comparision. All thesestatistics demonstrate that linearly interpolating thedBZ values is better. The total number of triadsconsidered and the finite values of the variances makemost statistical tests here superfluous. For example,the significant t-value for a paired difference testat 108 degrees of freedom and significance level of0.001 is about 3.3 whereas the computed t-value is onthe order of 104. Thus the differences in this tableare all statistically significant.

Do the results change if one were to consideronly low reflectivity values, only high reflectivityvalues, or only triads where the endpoints havelarge reflectivity gradients? The answer to all ofthese questions, as shown in the remaining rows ofTable 1, is “no”. Whether one considers the overallset of triads, or slices and dices them based on thereflectivity values or based on reflectivity gradients,

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Table 1: Comparing two forms of interpolation. The first row shows the statistics over the complete year ofdata. The remaining rows show statistics when the data are divided in different ways based on the values of theendpoints of the triads: cases where one endpoint is greater than some threshold, cases where both endpoints aregreater than some threshold and cases where the gradient between the endpoints is greater than some threshold.The penultimate column is the fraction of triads where interpolating in dBZ is better (ties are not counted) and thelast column provides the total number of triads in each category.

Triad Category Interpolating Z Interpolating dBZ Fraction where Number ofRMSE (dB) RMSE (Z) RMSE (dB) RMSE (Z) dBZ better triads

overall 4.17 6795.74 3.8 6713.8 0.54 2.2e+09one > 0 dBZ 4.08 7234.26 3.65 7147.04 0.55 2.0e+09

one > 20 dBZ 4.29 16962 3.44 16757.5 0.57 3.6e+08one > 40 dBZ 4.63 62600.1 3.66 61835.2 0.55 2.6e+07both > 0 dBZ 3.49 7840.36 3.26 7747.59 0.55 1.7e+09both > 20 dBZ 2.78 20480.6 2.64 20356.9 0.54 2.4e+08both > 40 dBZ 2.77 91436.3 2.73 93215.5 0.52 1.0e+07

gradient > 2.5 dBZ/km 6.35 10218.4 5.33 9848.66 0.6 5.4e+08gradient > 5 dBZ/km 9.15 14129.8 6.99 12656.3 0.65 1.6e+08

the result is consistent: interpolating in dBZ resultsin a lower interpolation error. Physically, then, thequantity being measured (the hydrometeor contentof the atmosphere) by a weather radar is morelinear in dBZ than in Z. Hence, based on the data,interpolation across range gates is, on average, moreaccurate when done in dBZ than in Z.

The gradient-wise segregations point outsomething very critical (Figure 1). Based onthe statistical evidence, as the absolute value of thegradient (between the two values being interpolatedover) increases, interpolation in dBZ becomes theclearly better choice. Indeed, if the two end pointsare separated by more than 10 dBZ (so that thegradient is more than 5 dBZ/km), interpolation indBZ is better twice as often (65% to 35%) thaninterpolation in Z. The higher the gradient, i.e. theless uniform the field, the more important it is thatinterpolation be carried out in dBZ.

4. Discussion

The implications of this result go beyond justinterpolation and play into the choice of howto carry out any linear operations on the radarreflectivity values at adjacent gates. Nearly allimage processing filtering operations are linear. Forexample, smoothing a field is commonly done byreplacing a pixel by the average value of its neighbors.This is the same as subtracting an error estimate from

Figure 1: The odds favoring dBZ interpolationover interpolation in Z as the gradient exceeds aspecific threshold. The higher the gradient, the moreimportant it is that interpolation be carried out indBZ.

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each pixel’s value, with the error estimate being x−x̄,where x is the value at that pixel and x̄ the averagepixel value in the pixel’s neighborhood. Expand outx̄ in terms of the neighboring values and it is easyto see that smoothing a dBZ field implies subtractinga fraction of the dBZ values of neighboring pixels.Hence, smoothing of radar reflectivity images is betterdone in dBZ than in Z.

The literature on image processing of weatherradar data has gotten this right, carrying outprocessing of adjacent range gates in dBZ andnot Z. A median filter is different because x̄ isindependent of monotonic transformations in x, buttechniques that use a weighted average (Delanoyand Troxel 1993; Lakshmanan 2004; Anagnostou2004) to compute x̄ involve subtracting dBZ values.Similarly, parameters such as the texture of thedBZ field used by Kessinger et al. (2003) boildown to arithmetic operations on dBZ values, notZ. Methods of storm identification from reflectivityalso operate on constant-dBZ increments (Johnsonet al. 1998; Rosenfeld 1987; Crane 1979) becauseincrements in Z are numerically problematic. Suchconstant dBZ increments, not constant Z increments,are widely used whenever adjacent pixel values needto be compared, for example by Steiner and Smith(2002) to compute their textural feature.

Pixel-wise operations such as Z-R relationshipscan afford to operate in Z, but any clustering,smoothing, segmentation or feature extractiontechnique that relies on the reflectivity value atadjacent pixels is better done in dBZ than in Z.

ACKNOWLEDGMENTS

Funding for the author was provided byNOAA/Office of Oceanic and Atmospheric Researchunder NOAA-University of Oklahoma CooperativeAgreement #NA11OAR4320072, U.S. Departmentof Commerce. I wish to thank John Cintineo forproviding the year’s archive of data, and acknowledgehelpful discussions with Kurt Hondl and RodgerBrown of NSSL.

REFERENCES

Anagnostou, E., 2004: A convective/stratiformprecipitation classification algorithm for volume

scanning weather radar observations. Meteor.Appl., 11, 291–300.

Charalampidis, D., T. Kasparis, and W. Jones, 2002:Removal of nonprecipitation echoes in weatherradar using multifractals and intensity. IEEE Trans.Geosci. Remote Sensing, 40, 1121–1132.

Crane, R., 1979: Automatic cell detection andtracking. IEEE Trans. Geosci. Electronics, 17,250–262.

Delanoy, R. and S. W. Troxel, 1993: Machineintelligent gust front detection. Lincoln LaboratoryJ., 6, 187–212.

Johnson, J., P. MacKeen, A. Witt, E. Mitchell,G. Stumpf, M. Eilts, and K. Thomas, 1998:The storm cell identification and trackingalgorithm: An enhanced WSR-88D algorithm.Wea. Forecasting, 13, 263–276.

Kessinger, C., S. Ellis, and J. Van Andel, 2003: Theradar echo classifier: A fuzzy logic algorithm forthe WSR-88D. 3rd Conf. on Artificial Applicationsto the Environmental Sciences, Long Beach, CA,Amer. Meteor. Soc., P1.6.

Lakshmanan, V., 2004: A separable filter fordirectional smoothing. IEEE Geosci. RemoteSensing Letters, 1 (3), 192–195.

Rosenfeld, D., 1987: Objective method for analysisand tracking of convective cells as seen by radar. J.Atmos. Oceanic Technol., 4, 422–434.

Steiner, M. and J. Smith, 2002: Use ofthree-dimensional reflectivity structurefor automated detection and removal ofnon-precipitating echoes in radar data. J. Atmos.Oceanic. Technol., 19, 673–686.

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REVIEWER COMMENTS

[Author’s responses in blue italics.]

REVIEWER A (Mark Askelson):

Initial Review:

Recommendation: Major revision

Substantive comments:

p1, P2: I believe that this is a good point, but note that, as indicated by the questions at the end of this paragraph,it is based upon the assumption that a linear interpolation technique is going to be used to estimate the valueat the missing pixel. Of course, this is only one objective analysis approach that one may use and oftentimeswill produce undesirable results (e.g., discontinuities in derivatives at observation locations). Moreover, objectiveanalysis fields commonly replicate nonlinear characteristics of input fields, which is, of course, desired. Thus,please consider either indicating earlier in the paragraph that the estimation method being used in this example islinear interpolation or making the argument more general.

Yes, I have now explicitly used “linearly interpolating” so that it is clear where I am headed. Also, as statedlater in this review, the point of this paper is not objective analysis but image processing. Thus, we are concernedmainly with linear operations over short distances. I use interpolation as the machinery to demonstrate linearity.

p1, P4: This is an important point and I am glad you have included it. However, I think it would be helpfulif you show how the two means are related so that the reader can have a better understanding of the arithmetic-vs. geometric-mean statement (here, capital letters indicate logarithmic units and lower-case letters indicate linearunitsplease see technical item 1 for why I follow this convention):

A1 =Z1 + Z2

2=

10 log z1 + 10 log z2

2=

10 log(z1z2)2

= 10 log(z1z2)1/2

and

A2 = 10 log a2 = 10 logz1 + z2

2

Thus, averaging two logarithmic radar reflectivity factors produces 10log of the geometric mean of the linear-unitsvalues while averaging of linear-units values, once converted into a result having logarithmic values, produces10log of the arithmetic mean. From a theoretical standpoint I believe that it is important to show this as itillustrates why the two approaches will not generally produce the same value.

Done.

In addition, I believe that the 52 dBZ result presented is incorrect. When I compute the average radar reflectivityvalue by converting 46 dBZ and 53 dBZ into linear values, averaging them arithmetically, and then converting theresult back into logarithmic values, I obtain 50.78 dBZ. Thus, I obtain a difference of 1.28 dB (the proper unitfor dBZ - dBZ is dB).

The reviewer is correct; I have now fixed this.

I have a concern here — it is that testing using logarithmic units inherently favors the arithmetic logarithmicaverage. I believe that to be fair, one should also compute errors in linear units in the following way, with ea log a

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meaning error of the arithmetic logarithmic average and elina meaning error of the arithmetic linear average:

ea log a(mm6mm−3) = z0 − 10z−1−z1

2 /10 = z0 − (z−1z1)1/2

andealina(mm6mm−3) = z0 −

z−1 + z1

2where again capital letters imply logarithmic units and lower-case letters imply linear units.

My concern regards the logarithmic operations that occur in different relative locations in (1) and (2) and whatimpact they have on relative error values. Putting (1) and (2) into linear units results in

ea log1=

z0

(z−1z1)1/2=

11 + ε1/z0

andealin2 =

z0z−1+z1

2

=1

1 + ε2/z0

where (z−1z1)1/2 = z0 + ε1 and z−1+z12 = z0 + ε2. If logarithmic values of radar reflectivity factor have a

linear spatial variation over three consecutive range resolution volumes, then ea log a = 0 and ea log a(1) = 1.Alternatively, if linear values of reflectivity factor have a linear spatial variation over three consecutive rangeresolution volumes, then ealina = 0 and ealin(2) = 1. Thus ea log a, ealina, ea log a(1), and ealin(2) providethe desired error characteristics for the situation of purely (spatially) linear fields (for ea log a(1) and ealin(2) thelogarithmic operation produces 0 dB in both cases).

However, because the error values computed using (1) and (2) involve taking the logarithm of ealoga(1) andealina(2), ε1 (or ε2) values that have the same magnitude (in linear units) but opposite signs will produce errorsthat have different magnitudes. If, for instance, ε1/z0 = +/-0.1, one obtains -0.4139 dB from (1) for ε1/z0 =+0.1 and 0.45757 dB for ε1/z0 = -0.1. Moreover, considering that the geometric mean is always smaller than thearithmetic mean and that arithmetically averaging logarithmic reflectivity values is equivalent to taking 10 log of thegeometric mean of the linear-units values whereas the other approach involves taking 10 log of the arithmetic meanof the linear-units values, it seems possible that ε1/z0 is slightly negative more often than ε2/z0 is which, owingto the logarithmic operation could skew the results. This argument, of course, leverages the idea of linear-unitsreflectivity values that are of the same magnitude but different signs, which of course will not generally be the case.However, I believe that it does illustrate the potential impact of the logarithmic operations in (1) and (2) and, thus,motivates the computation of errors using ea log a and ealina for, at the very least, comparison purposes.

Fair point: a new column has been added to the table, with the RMSE measured in mm3 mm−6. The resultsremain the same. In any area with some structure, the dBZ values are more linear. Also, the pair-wise tests (thefraction when dBZ is better) is independent of the units being used.

My most significant concern regards the linearity-implies-which-field-to-use-argument. While I agree that if theunderlying field is spatially linear one would do well to apply an analysis technique that retains its spatial linearity, Iargue that the situation is much more complicated than this. The examples given (linear interpolation and boxcaraveraging) are both relatively simple objective analysis schemes, the first of which enforces linearity betweeninterpolation points and the second of which does not generally produce spatially-linear analysis fields (andboth of which have significant shortcomings like discontinuities in derivative fields and ringing in the responsefunction). Objective analysis schemes used in the atmospheric sciences can, generally, be expressed in termsof Distance-Dependant Weighted Averaging (DDWA; e.g. Askelson et al. 2005), which is a linear operation.However, just because DDWA is a linear operation, this does not mean that DDWA will produce linear spatial(or temporal) variations. On the contrary, most geophysical fields contain significant nonlinearity, the degreeof which depends upon the scales being considered. Thus, DDWA schemes generally retain at least some ofthis nonlinearity. Radar reflectivity factor fields, whether viewed using logarithmic or linear units, undoubtedly

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contain nonlinear variations. Here, values were analysed over a 3 km distance. Of course, even a highlynonlinear field can be decomposed into linear changes if these linear changes occur over small enough distances.

The finding that logarithmic-units values are more linear over small distances (up to 3 km) than linear-units valuesonly, in my opinion, has relevance for variations having scales of up to 3 km. Because DDWA schemes arecommonly applied over distances greater than 3 km when analysing radar data and because they may not retainspatial linearity on a 3 km scale, I believe that statements should be qualified in a manner similar to “. . . if one isusing an objective analysis method that retains spatial linearity over short distances, then using logarithmic valuesis preferable relative to using linear values.” Thus, my concern is that given the evidence presented here, thestatement that logarithmic reflectivity values should be used in all objective analysis schemes (all smoothing) is toobroad. Before closing it is noted that if a field is linear over certain spatial scales and an objective analysis does notreplicate that linearity, then this would of course be a detriment. However, such a scheme may sacrifice linearityon those scales to more accurately represent variations on other scales. Consequently, trade-offs at different scalesmay occur that may be good or bad, depending upon an analysts preferences.

The goal of the article is to determine the field on which image processing operations ought to be carried out. It isnot about determining how to carry out objective analysis. In other words, we know that most image processingoperations (e.g: interpolation, smoothing, clustering) are linear and work only when the data values that arebeing operated on vary linearly. I use linear interpolation at known values to see if the field varies linearly overthe scale of a few pixels. It turns out that operating in dBZ is better than operating in Z if we want local linearity.Therefore, I argue, it is better to carry out image processing in dBZ.

The reviewer is absolutely correct that over larger distances, atmospheric fields are non-linear. But that does notaffect our argument which is solely about image processing operations (not objective analysis).

There are two reasons why such nonlinearity over large distances does not affect our argument (again our argumentis about image processing, not objective analysis: the title of the paper makes our focus quite clear). Firstly,linear interpolation is used in this paper just as the machinery to verify a Hilbert space — a system of unitswithin which spatial variation at the scale of a few image pixels is linear. Thus, we are not concerned with linearinterpolation over large distances or multiple pixels. Secondly, most image processing operations (smoothing,texture, subsampling, etc.) model an image as a Markov random field by which we mean that they assume apixel’s value is correlated only on its immediate neighbors. Thus, non-linear, sharp spatial changes will be poorlyhandled anyway by image processing operations. This is regardless of what space we operate in.

I think it is clearer to refer to reflectivity factor, regardless of the units, as reflectivity factor and then to specificallyindicate whether its units are linear or logarithmic. Rinehart (2010) keeps these distinct by using a lower case zto indicate that the radar reflectivity factor units are linear and an upper case Z to indicate that the radar reflectivityfactor units are logarithmic. While I am not suggesting that you have to do this, the use of the symbol dBZ toindicate logarithmic units is a bit awkward, in my opinion, since dBZ indicates units, not the variable itself. Onewould not, for instance, label an east-west distance variable as ft.

This is something I went back-and-forth on. I decided to use the terminology in this paper because it is muchclearer (when spoken, for example) than the z-Z dichotomy. Somewhat equivalently, we could talk about workingin standard-atmospheres or hPa and I believe most people would understand whether we are working with ratiosor millibars.

REVIEWER B (Mario Majcen):

Initial Review:

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Recommendation: Accept with minor revisions

General comments:

The paper is acceptable with minor revisions and no further review is requested unless major changes are made inaccordance with other reviews. I really enjoyed reading this manuscript. It reveals a fallacy behind a commonlyused data manipulation technique.

REVIEWER C (David Dowell):

Initial Review:

Recommendation: Accept with minor revisions

General comments: This short, focused article summarizes results of an important study for radar-data analysis.The writing style is somewhat informal, which mostly works for this article. A few instances where more preciseand/or formal wording would help are mentioned below. Also, some suggestions are provided to avoid overstatingresults, resulting in a stronger manuscript.

Substantive comments: In paragraph 2, the motivation for the study relies on hearsay. Can we instead refer tomore concrete evidence (preferably publications) that the community often favors Z processing over dBZ?

As stated in the manuscript, the literature on the topic has it right — published work in this area involves processingdBZ, not Z. I would hypothesize that even believers in Z quickly discover that it doesn’t work well, and resort toworking in dBZ. Thus, I was unable to find any publication doing anything in Z.

p. 3, column 2, paragraph 1: If retained, the discussion This is the same neighboring pixels. belongs earlier inthe manuscript and requires more explanation. Or maybe this discussion can be deleted entirely.

This is an important point. I’ve retitled this section “Discussion” and added the link text that we are going beyondlinear interpolation to addressing any linear filtering operation.

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